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# Topics in Probability: The mathematics of financial risk-management

## Instructor: Marco Avellaneda

## Courant Institute of Mathematical Sciences

## Spring, 1996

(Initial blurb (1996))
This course is divided into four parts. First, we
will cover fundamentals on Ito Calculus and Arbitrage Pricing Theory,
emphasizing the use of trinomial trees, or finite-difference schemes,
for pricing derivative securities
(options, forwards, callable bonds,
etc.) Next, we will move into the area of transaction costs due
to liquidity constraints. We will analyze how low liquidity
in trading the underlying asset
affects implied volatility and dynamic hedging.
Third, we will consider different ways of
of modeling markets with
uncertain volatility. We will cover stochastic
volatility models, auto-regressive models as well as recently derived
nonlinear pricing models.
We will emphasize, in particular,
worst-case scenario risk-management techniques, and managing
volatility risk with options.
Finally, we
will cover more practical risk-management
techniques and guidelines
such as
the
``Value at Risk'' outlined in the J.P. Morgan
technical document on measuring financial risk.

# Mathematics of Finance II

(Spring 1998) Mathematics of Finance II borrows some of the same basic
material from the original Risk-management
course but emphasizes mostly (i) continous time finance
and (ii) modeling the term structure of interest rates.
We do a little bit of ``taxonomy'' of models as well as
some conceptual thinking about the significance and applications
of term-structure models.
Topics tend to change from one year to the next.

## Lecture Notes

1. Syllabus.

2. Brownian Motion and Ito Calculus.

3. Ito processes, continuous-time martingales
and Girsanov's Theorem (revised)

4. Continuous-time finance: an introduction

5. Valuation of derivative securities

6. Uncertain Volatility Model & worst-case
scenario pricing

.ps file of transparencies. For more complete notes, look at the
first two papers

in
``Recent papers in Mathematical Finance''.

7. Trinomial trees and finite-difference schemes

## INTEREST RATE DERIVATIVES (Spring 1998, very rough drafts)

8. Basic term structure concepts

9. Heath-Jarrow-Morton thm and forward rate correlations

10. Affine term structure models

## NEW PROBLEMS (Spring 1998)

#1 Brownian exit time from a strip

(2nd corrected version: thanks
to those that pointed out earlier mistakes! :-) )
#2 Gauss-Markov processes

#3 Term-structure Modeling

## Old Homework Assignments

Homework #1 (revised)

Extra problem: digital
barrier and volalility term-structure

Homework # 2